Question

A scuba diver's tank contains \(0.29 \mathrm{~kg}\) of \(\mathrm{O}_{2}\) compressed into a volume of \(2.3\) L. (a) Calculate the gas pressure inside the tank at \(9{ }^{\circ} \mathrm{C}\) (b) What volume would this oxygen occupy at \(26^{\circ} \mathrm{C}\) and \(0.95 \mathrm{~atm} ?\)

Short answer

(a) The gas pressure inside the tank at $9^\circ{C}$ is \(8.48~\mathrm{atm}\). (b) The volume the oxygen would occupy at $26^\circ{C}$ and $0.95 ~\mathrm{atm}$ is \(23.25~\mathrm{m^3}\).

Step-by-step solution

Calculate the number of moles

To find the number of moles, we can use the formula: n = mass / molar mass Given the mass of oxygen in the tank is 0.29 kg, we need to find the molar mass of oxygen gas (O₂). The atomic mass of an oxygen atom is 16 g/mol, and since there are 2 oxygen atoms, the molar mass of oxygen gas is 32 g/mol. Let's convert the mass of oxygen gas from kg to grams: mass = 0.29 kg × 1000 g/kg = 290 g Now, we can find the number of moles: n = 290 g ÷ 32 g/mol = 9.0625 mol

Convert temperature to Kelvin

We are given that the temperature inside the tank is 9°C. To use Ideal Gas Law, we need to convert the temperature from Celsius to Kelvin using the formula: T(K) = T(°C) + 273.15 T(K) = 9°C + 273.15 = 282.15 K

Use Ideal Gas Law to find pressure

Now, we'll use Ideal Gas Law (PV = nRT) to find the pressure inside the tank, where R is the gas constant 8.314 J/(mol·K). We are given the volume in liters, so let's first convert that to cubic meters: V = 2.3 L × 0.001 m³/L = 0.0023 m³ Now, we can solve for the pressure (P): P = nRT / V P = (9.0625 mol × 8.314 J/(mol·K) × 282.15 K) / 0.0023 m³ P = 859077.628 J/m³ Since 1 Pa = 1 J/m³, let's convert the pressure to atm by dividing by 101325 Pa/atm: P = 859077.628 J/m³ ÷ 101325 Pa/atm = 8.48 atm For part (a), the gas pressure inside the tank at 9°C is 8.48 atm. Now, for part (b), we'll find the volume the oxygen would occupy at 26°C and 0.95 atm.

Convert the new temperature to Kelvin

We are given that the new temperature would be 26°C. Let's convert the temperature from Celsius to Kelvin using the formula: T(K) = T(°C) + 273.15 T(K) = 26°C + 273.15 = 299.15 K

Use Ideal Gas Law to find the new volume

We already know the number of moles (n) and the new temperature (T) and pressure (P). Using Ideal Gas Law, we can find the new volume (V) of oxygen gas occupying the space: V = nRT / P V = (9.0625 mol × 8.314 J/(mol·K) × 299.15 K) / (0.95 atm × 101325 Pa/atm) V = 2241411.516 J/m³ ÷ 96308.75 J/m³ V = 23.25 m³ For part (b), the volume the oxygen would occupy at 26°C and 0.95 atm is 23.25 m³.

Key concepts

Gas Pressure Calculation

Understanding how to calculate gas pressure is essential in studying the behavior of gases.
The Ideal Gas Law formula, given as \(PV = nRT\), allows us to calculate the pressure (\(P\)) if we know the volume (\(V\)), number of moles (\(n\)), temperature in Kelvin (\(T\)), and the ideal gas constant (\(R\)).

Calculating Pressure

To find the pressure of gas, as in our scuba diver's tank example, you rearrange the Ideal Gas Law to solve for \(P\): \(P = \frac{nRT}{V}\). It is crucial to ensure all measurements are in their proper units: moles for the quantity of gas, Kelvin for temperature, and liters or cubic meters for volume.

Conversion to Standard Units

In the scuba tank scenario, the number of moles was calculated, the temperature was converted to Kelvin, and volume was given in liters, which we then converted to cubic meters to match the standard unit for pressure calculations in J/m³ (Pascals).

Unit Conversion

Once calculated, pressure in Pascals can be converted to atmospheres (atm) by dividing by 101325 Pa/atm, which is more familiar in chemistry and everyday life. In the divers' tank example, we did this to express the pressure in atm, a unit commonly used to discuss gas pressure underwater.

Temperature Conversion

Temperature plays a pivotal role in gas behavior and pressure calculations.
To work with the Ideal Gas Law, the temperature must be in Kelvin, which is the SI unit for thermodynamic temperature scale.

From Celsius to Kelvin

The conversion from Celsius to Kelvin is straightforward: \(T(K) = T(^\circ C) + 273.15\). This adjustment accounts for the difference in the starting points of the two scales: 0 K is absolute zero, while 0°C is the freezing point of water.

Why Kelvin for Gases?

Kelvin is used because it allows for the calculation of gas properties consistently since temperature in Kelvin is directly proportional to the kinetic energy of the molecules. This consistency simplifies equations and computations in thermodynamics.

Moles of Gas

The mole is a foundational concept in chemistry and a primary unit in the Ideal Gas Law.
It relates the quantity of gas to its molecular nature, providing a bridge between the macroscopic world we observe and the microscopic world of atoms and molecules.

What is a Mole?

A mole is defined as the amount of substance that contains as many entities (atoms, molecules, etc.) as there are atoms in 12 grams of carbon-12. This number is Avogadro's number, approximately \(6.022 \times 10^{23}\) entities per mole.

Calculating Moles in Practice

In context, the number of moles (\(n\)) is typically found by dividing the mass of a substance by its molar mass. For gases, this helps determine how much gas is present. In the textbook problem, the weight of oxygen (in kilograms) had to be converted to grams before determining the moles of gas, using the molar mass of oxygen.

Molar Mass of Gas

Molar mass, the mass of one mole of a substance, is vital in quantifying gases and using the Ideal Gas Law.
It is measured in grams per mole (g/mol) and is numerically equivalent to the average mass of one molecule of the substance.

Finding Molar Mass

The molar mass of an elemental molecule (like \(O_2\)) is the sum of the atomic masses of all atoms in the molecule. For oxygen gas, with its diatomic molecules, you multiply the atomic mass of an oxygen atom (16 g/mol) by two.

Practical Application

In gas law calculations, the molar mass allows us to convert between the mass of a gas sample and the number of moles present, a necessary step before applying the Ideal Gas Law, as demonstrated when we calculated the moles of oxygen in the diver's tank.